Page:Carroll - Euclid and His Modern Rivals.djvu/193

. § 3.] Here, for instance, is something about 'Directives,' which seem to be a curious kind of Loci—quite different from Right Lines, I should say.

Nie. Oh no! They are exactly the same thing!

Min. Well, I find, at p. 4, 'Directives are either divergent or parallel': and again, at p. 11, 'Parallel Directives cannot meet.' Clearly, then, Directives can never by any possibility coincide: but ordinary Right Lines occasionally do so, do they not?

Nie. It is a curious lapsus pennae.

Min. At p. 7, I observe an article headed 'The principle of double conversion,' which I will quote entire.

Reads. 'If four magnitudes, a, b, A, B, are so related, that when a is greater than b, A is greater than B; and when a is equal to b, A is equal to B: then, conversely, when A is greater than B, a is greater than b; and, when A is equal to B, a is equal to b.

'The truth of this principle, which extends to every kind of magnitude, is thus made evident:—If, when A is greater than B, a is not greater than b, it must be either less than or equal to b. But it cannot be less; for, if it were, A should, by the antecedent part of the proposition, be less than B, which is contrary to the supposition made. Nor can it be equal to b; for, in that case, A should be equal to B, also contrary to supposition. Since, therefore, a is neither less than nor equal to b, it remains that it must be greater than b.' Now let a and A be variables and represent the ordinates